# Practical applications of the $L^p$ norm when $p \neq 1,2,\infty$

I'm roughly familiar with the concept of $L^p$ norms -- what they represent and how they are computed -- though I am far from educated in functional analysis in general. For reasons that are more or less obvious, in many practical applications, the $L^2$ norm is used (ie, optimization, etc). One also sees the $L^1$ and $L^{\infty}$ norms used from time to time.

However, rarely do I see any mention of, say, the $L^3$ norm, or really any $L^p$ norm where $p$ is something other than 1, 2 or $\infty$ (or, in some interpretations, 0).

Are there any algorithms or applications that have exploited the other norms? Does convergence in an $L^p$ sense mean something different for these values of $p$? Is there any value of $p$ that can lead to a property not obtainable by $p=1,2,\infty$?

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Relevant: mathoverflow.net/questions/28147/… – Jesse Madnick Sep 22 '12 at 3:43
The $L_0$-"norm", (a misnomer since it is not a norm in the usual sense), which counts the number of non-zero components of a vector, is a convenient notation to deal with sparsity in compressed sensing. – user17762 Sep 22 '12 at 4:21

## 1 Answer

One example not mentioned in the MO thread: the $L^4$ norm appears in the Ginzburg-Landau formula for the energy of superconductor.

More generally: we want nonlinear variational models to describe phenomena which do not obey linear superposition. Minimization of $L^2$ norm leads to linear PDE. Minimization of $L^1$ and $L^\infty$ norms leads to badly degenerate things that are barely PDE at all. The norms $L^4$ and $L^{\rm dimension}$ emerge as attractive alternatives.

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Along these lines, could seeking alternative norms for minimization be a general strategy for solving other problems? – Emily Sep 24 '12 at 18:15
@EdGorcenski One usually minimizes an energy functional, which is not necessarily a norm (i.e., neither homogeneity nor triangle inequality are required). But energy functionals tend to have bits of $L^p$ norms in them. A Google Scholar search shows that $L^2$ and $L^4$ are the most popular ingredients. – user31373 Sep 24 '12 at 18:51